MC-RFM: Geometry-Aware Few-Shot Adaptation via Mixed-Curvature Riemannian Flow Matching

Researchers introduce MC-RFM, a novel framework for efficiently adapting frozen vision models to new tasks using mixed-curvature Riemannian geometry. The method represents adapted features on a product manifold combining hyperbolic and Euclidean spaces, outperforming existing parameter-efficient adaptation techniques across multiple benchmarks and backbone architectures.

MC-RFM addresses a fundamental limitation in current few-shot adaptation methods: they treat feature adaptation as simple Euclidean perturbations without considering the geometric structure of task-induced feature displacement. This research demonstrates that explicitly modeling the mathematical geometry underlying feature transformations yields measurable performance improvements. The framework's use of mixed-curvature manifolds is particularly innovative—hyperbolic geometry naturally captures hierarchical semantic relationships while Euclidean geometry preserves local discriminative patterns, creating a geometrically appropriate representation space.

The approach emerges from growing recognition in machine learning that representation geometry matters as much as what parameters are updated. Traditional few-shot adaptation methods focus on parameter efficiency through linear probes, prompts, or low-rank updates, but overlook how representations should optimally move through feature space. MC-RFM frames adaptation as continuous transport along task-conditioned paths, fundamentally reconceptualizing the adaptation problem.

For AI practitioners and researchers, this work has immediate implications for deploying vision models across diverse domains with limited labeled data. The method's backbone-agnostic design and operation on frozen features mean seamless integration with existing pretrained models without architectural modifications. Performance gains on fine-grained datasets and Transformer backbones suggest particular value for modern architectures. The comprehensive ablation studies validate that mixed-curvature geometry, task conditioning, and adaptive gating all contribute meaningfully to performance improvements, providing a solid theoretical foundation.

Future developments may extend mixed-curvature approaches to other adaptation modalities, explore connections to neural flow models in vision, or investigate how geometric properties correlate with downstream task characteristics. This work opens possibilities for geometry-informed adaptation across language and multimodal domains.

• →MC-RFM uses mixed-curvature Riemannian manifolds to geometrically model feature adaptation in few-shot learning scenarios
• →The framework outperforms existing parameter-efficient methods on seven benchmarks across multiple frozen backbone architectures
• →Hyperbolic geometry captures semantic hierarchy while Euclidean geometry preserves local discriminative variation in the representation space
• →The method operates entirely on cached frozen features with no backbone modifications, enabling practical deployment
• →Strongest performance gains appear on Transformer backbones and fine-grained visual recognition datasets